Fine tuning the polynomial Model

Let’s address here a couple of the to-dos from the previous pages. Namely, let’s check the degree of the polynomial using a grid search and checkout other climatic variables, playing a bit with variable selection.

Tune polynomial degree

First, let’s fine-tune the degree of the polynomial. From the initial scatter plots, it seems that a 2 or 3 degree polynomial would be good. But let’s throw a grid search on that.

♻️  stepit 'grid_search_pipe': is up-to-date. Using cached result for `strom.modelling.grid_search_pipe()` 2025-11-22 16:19:31

/home/runner/work/strom/strom/src/strom/strom.py:837: FutureWarning:

Setting an item of incompatible dtype is deprecated and will raise in a future error of pandas. Value '['2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7'
 '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7' '2' '3' '4' '5' '6' '7']' has dtype incompatible with int64, please explicitly cast to a compatible dtype first.

MAE

MSE

RMSE

R2

MAPE

EVS

MeAE

D2

Pinball

It seems degree 4 would be best. Of course there is a a small variation according to the metric/scorer used, but overall, I think 4 seems best. The differences are rather small, but there is still this tension between MAE and RSME. A model that better fit on the warmer season and therefore has smaller small errors and a small MAE, tend to have a few very large errors, affecting the RSME. And the other way around.

Variable selection

Now let’s try here a brute-force approach to variable selection. Thereby, taking a not-so-thoughtful-but-quick-and-effective approach to trying out other climatic variables and checking the associations with relative humidity and other variables, to see if there is indeed a signal or just noise in some of them like humidity and so on.

Just let it crunch through a bunch of variable combinations. There will be many non-sensical or irrelevant combinations. But it’s just fast to write and the machine will have to work, me not so much.

From topic knowledge and the first correlations observed, we would expect mostly temprature to play a key role in the model. Yet, other variables such a humidity, pressure or condensation point could also be relevant. So let’s throw all that into a grid search and see what it spits out of it.

• tt: Temperatur der Luft in 2m Hoehe °C
• rf_tu: relative Feuchte %
• td: Taupunktstemperatur °C
• vp_std: berechnete Stundenwerte des Dampfdruckes hpa
• tf_std: berechnete Stundenwerte der Feuchttemperatur °C
• p_std: Stundenwerte Luftdruck hpa

♻️  stepit 'grid_search_pipe': is up-to-date. Using cached result for `strom.modelling.grid_search_pipe()` 2025-11-22 16:19:32

{'vars__columns': ['tt_tu_mean', 'vp_std_mean', 'tf_std_mean']}

MAE

MSE

RMSE

R2

MAPE

EVS

MeAE

D2

Pinball

Loading ITables v2.5.2 from the init_notebook_mode cell... (need help?)

Well humidity and other climatic variables can only improve the model marginally. A proper mediation analysis would still be in order, but this at least shed some light on it. Interestingly, some models without temperature but the set of other climatic variables almost equal the performance of the best model with temperature. Overall, it seems that at least temperature, humidity and pressure should be considered. Yet, the tend to be collineal and thus not really be able to used them all just like that in this kind of model.

One last brute-force approach for today and let automatically choose the best model

Best parameters

{'polynomial__degree': 4, 'vars__columns': ['tf_std_mean']}

Pipeline(steps=[('vars', ColumnSelector(columns=['tf_std_mean'])),
                ('polynomial', PolynomialFeatures(degree=4)),
                ('model', LinearRegression())])

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/home/runner/work/strom/strom/src/strom/strom.py:837: FutureWarning:

Setting an item of incompatible dtype is deprecated and will raise in a future error of pandas. Value '['3' '3' '3' ... '5' '5' '5']' has dtype incompatible with int64, please explicitly cast to a compatible dtype first.

Loading ITables v2.5.2 from the init_notebook_mode cell... (need help?)

Loading ITables v2.5.2 from the init_notebook_mode cell... (need help?)

Assessment of the best model in that brute force approach

♻️  stepit 'poly': is up-to-date. Using cached result for `strom.modelling.assess_model()` 2025-11-22 16:19:33

Metrics

Plot

Table

Loading ITables v2.5.2 from the init_notebook_mode cell... (need help?)

Scatter plot matrix

Test

Train

Observed vs. Predicted and Residuals vs. Predicted

Check for …

check the residuals to assess the goodness of fit.

• white noise or is there a pattern?
• heteroscedasticity?
• non-linearity?

Test

Train

Normality of Residuals:

Check for …

• Are residuals normally distributed?

Test

Train

Test

Train

Leverage

Test

Train

Scale-Location plot

Test

Train

Residuals Autocorrelation Plot

Test

Train

Residuals vs Time

Test

Train

Compare models

So let’s compare the models with and without humidity.

Cross-validation messages

♻️  stepit 'cross_validate_pipe': is up-to-date. Using cached result for `strom.modelling.cross_validate_pipe()` 2025-11-22 16:19:37

♻️  stepit 'cross_validate_pipe': is up-to-date. Using cached result for `strom.modelling.cross_validate_pipe()` 2025-11-22 16:19:37

♻️  stepit 'cross_validate_pipe': is up-to-date. Using cached result for `strom.modelling.cross_validate_pipe()` 2025-11-22 16:19:37

Metrics

Single split

Metrics based on the test set of the single split

Line plot

Radar plot

Cross validation

Line plot

Radar plot

MAE

MSE

RMSE

R2

MAPE

EVS

MeAE

D2

Pinball

Predictions, residuals, observed

next

Time vs. Predicted and Observed

Time vs. Residuals

Model details

2c715

Pipeline(steps=[('vars', ColumnSelector(columns=['tt_tu_mean', 'rf_tu_mean'])),
                ('polynomial', PolynomialFeatures()),
                ('model', LinearRegression())])

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77de0

Pipeline(steps=[('vars', ColumnSelector(columns=['tt_tu_mean', 'tf_std_mean'])),
                ('polynomial', PolynomialFeatures(degree=4)),
                ('model', LinearRegression())])

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c20ad

Pipeline(steps=[('vars', ColumnSelector(columns=['tf_std_mean'])),
                ('polynomial', PolynomialFeatures(degree=4)),
                ('model', LinearRegression())])

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A nice improvement over the baseline model, and the model without humidity is slightly better across all metrics (a rather small improvement, but across all metrics).

TODOs

• Let’s just compare the models diretly.
• Conduct a proper mediation analysis to tease out the relationship between the climatic variables and wärmestrom consumption
• Already mentioned in the previous model, but the results here reinforce that it would be a good idea to use PCA for feature engineering and make the most out of the -highly correlated- climatic variables
• Create interaction terms between important features
• Derive new features based on domain knowledge (e.g., day of week, holidays)
• Improve hyperparameter tuning, for example, using randomized grid search to explore a wider range of parameters more efficiently, and possibly also using more advanced strategies like Bayesian optimization (scikit-optimize) or genetic algorithms.
• try lagged features and incoporate seasonal-trend decomposition in the model.
• after that kind of feature engineering, use perhaps recursive feature elimination instead of the brute-force approach.